This is a SYSTEM of differential equations that you can rewrite as:
[tex]\frac{d\vec{U}}{dt}=A\vec{U}, \vec{U}=\frac{d\vec{R}}{dt}[/tex]
and A is a matrix.
If A has constant coefficients, then the system is readily solvable with eigen-vector decomposition.

Question:
Did you get this equation from a physical problem with a Coriolis term?
Just curious..

EDIT:
Insofar as B is constant, you may gain two decoupled 3.order diff.eq's in R (i.e 2.order in U).

You can, as Arildno suggested, introduce [itex]u= \frac{dx}{dt}[/itex] and [itex]v= \frac{dy}{dt}[/itex] and write this as a system of 4 first order differential equations.

Another way to handle it is this: differentiate the first equation again to get
[tex]\frac{d^3x}{dt^3}+ B\frac{d^2y}{dt^2}= 0[/itex]
and use the second equation to substitute for the second derivative of y
[tex]\frac{d^3x}{dt^3}+ B^2\frac{dx}{dt}= 0[/tex]
That's easy to solve.

Once you know x(t), you can use the first equation to solve for [itex]\frac{dy}{dt}[/itex] and integrate once more to find y(t).

(I just noticed J77's comment. I am assuming here that B is a constant.)